
A famous theorem of Kisin [1]} indicates that those modules are in
correspondence with \(\overline{{\mathbb {Z}}}_p\) -representations of \(G_K\)  that become
crystalline over the extension \(K[\@root e \of {p}]\) .
Besides, the Hodge type (resp. the inertial type) of the latter can be
easily read off on the form of the Frobenius map (resp. of the descent
data) on the former.
Even better, the reduction modulo \(p\)  of the representation associated
to a Breuil-Kisin module \(\mathfrak {M}\)  is uniquely and entirely described by
the module \(\mathfrak {M}\otimes _{{\mathcal {O}}_K[[u]]} k_K((u))\)  equipped with its
additional structures.
